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Generalized moments of sums with heavy-tailed random summands

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Abstract

In this paper, we investigate the asymptotic behavior of randomly weighted sums of the form of \({S}_{n}^{\theta \xi }={\theta }_{1}{\xi }_{1}+\cdots +{\theta }_{n}{\xi }_{n}\) under the transformation φ: ℝ → ℝ satisfying several asymptotic properties. The collection {ξ1,…,ξn} consists of dominatedly varying, not necessarily identically distributed, random variables following a specific dependence structure, whereas {θ1,…,θn} comprise of possibly dependent nonnegative and nondegenerate at zero random weights. Both collections are assumed to be independent. Inspired by the recent results. we obtain asymptotic bounds for the tail expectation \(\mathbf{E}\left(\varphi \left({S}_{n}^{\theta \xi }\right){1}_{\left\{{S}_{n}^{\theta \xi }>x\right\}}\right)\) expressing them by the sums of tail expectations \(\mathbf{E}\left(\varphi \left({\theta }_{k}{\xi }_{k}\right){1}_{\left\{{\theta }_{k}{\xi }_{k}>x\right\}}\right).\) We provide several particular examples to illustrate the obtained results.

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  1. Institute of Mathematics, Vilnius University, Naugarduko str. 24, LT-03225, Vilnius, Lithuania

    Mantas Dirma, Neda Nakliuda & Jonas Šiaulys

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  1. Mantas Dirma
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  2. Neda Nakliuda
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  3. Jonas Šiaulys
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Dirma, M., Nakliuda, N. & Šiaulys, J. Generalized moments of sums with heavy-tailed random summands. Lith Math J 63, 254–271 (2023). https://doi.org/10.1007/s10986-023-09606-y

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